A convergence technique for the game i-Mark
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arXiv
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| Format: | Preprint |
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2025
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| author | Nivasch, Gabriel Rubinstein, Oz |
| author_facet | Nivasch, Gabriel Rubinstein, Oz |
| contents | The game of i-Mark is an impartial combinatorial game introduced by Sopena (2016). The game is parametrized by two sets of positive integers $S$, $D$, where $\min D\ge 2$. From position $n\ge 0$ one can move to any position $n-s$, $s\in S$, as long as $n-s\ge 0$, as well as to any position $n/d$, $d\in D$, as long as $n>0$ and $d$ divides $n$. The game ends when no more moves are possible, and the last player to move is the winner. Sopena, and subsequently Friman and Nivasch (2021), characterized the Sprague-Grundy sequences of many cases of i-Mark$(S,D)$ with $|D|=1$. Friman and Nivasch also obtained some partial results for the case i-Mark$(\{1\},\{2,3\})$.
In this paper we present a convergence technique that gives polynomial-time algorithms for the Sprague-Grundy sequence of many instances of i-Mark with $|D|>1$. In particular, we prove our technique works for all games i-Mark$(\{1\},\{d_1,d_2\})$.
Keywords: Combinatorial game, impartial game, Sprague-Grundy function, convergence, dynamic programming. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_23196 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A convergence technique for the game i-Mark Nivasch, Gabriel Rubinstein, Oz Combinatorics Discrete Mathematics 91A46 The game of i-Mark is an impartial combinatorial game introduced by Sopena (2016). The game is parametrized by two sets of positive integers $S$, $D$, where $\min D\ge 2$. From position $n\ge 0$ one can move to any position $n-s$, $s\in S$, as long as $n-s\ge 0$, as well as to any position $n/d$, $d\in D$, as long as $n>0$ and $d$ divides $n$. The game ends when no more moves are possible, and the last player to move is the winner. Sopena, and subsequently Friman and Nivasch (2021), characterized the Sprague-Grundy sequences of many cases of i-Mark$(S,D)$ with $|D|=1$. Friman and Nivasch also obtained some partial results for the case i-Mark$(\{1\},\{2,3\})$. In this paper we present a convergence technique that gives polynomial-time algorithms for the Sprague-Grundy sequence of many instances of i-Mark with $|D|>1$. In particular, we prove our technique works for all games i-Mark$(\{1\},\{d_1,d_2\})$. Keywords: Combinatorial game, impartial game, Sprague-Grundy function, convergence, dynamic programming. |
| title | A convergence technique for the game i-Mark |
| topic | Combinatorics Discrete Mathematics 91A46 |
| url | https://arxiv.org/abs/2503.23196 |